If g(x) is a continuously differentiable function, then to solve
∫ f(g(x)) g'(x) dx; we substitute g(x) = t and g'(x) dx will be equal to dt.
Hence the problem is transformed to ∫ f(t) dt
Example I = ∫tan x dx = ∫(sin x/cos x)dx
If f(x) = t, f'(x)dx = dt
cos x = t;
-sin x dx = dt
sin x dx = -dt
I = ∫(sin x/cos x)dx = ∫-dt/t = -log |t|+c = - log|cos x|+C
= log |sec x|+C